64,664 research outputs found
Hamilton's theory of turns revisited
We present a new approach to Hamilton's theory of turns for the groups
SO(3) and SU(2) which renders their properties, in particular their
composition law, nearly trivial and immediately evident upon inspection.
We show that the entire construction can be based on binary rotations rather
than mirror reflections.Comment: 7 pages, 4 figure
Hamilton's Turns for the Lorentz Group
Hamilton in the course of his studies on quaternions came up with an elegant
geometric picture for the group SU(2). In this picture the group elements are
represented by ``turns'', which are equivalence classes of directed great
circle arcs on the unit sphere , in such a manner that the rule for
composition of group elements takes the form of the familiar parallelogram law
for the Euclidean translation group. It is only recently that this construction
has been generalized to the simplest noncompact group , the double cover of SO(2,1). The present work develops a theory of
turns for , the double and universal cover of SO(3,1) and ,
rendering a geometric representation in the spirit of Hamilton available for
all low dimensional semisimple Lie groups of interest in physics. The geometric
construction is illustrated through application to polar decomposition, and to
the composition of Lorentz boosts and the resulting Wigner or Thomas rotation.Comment: 13 pages, Late
Wigner distributions for finite state systems without redundant phase point operators
We set up Wigner distributions for state quantum systems following a
Dirac inspired approach. In contrast to much of the work on this case,
requiring a phase space, particularly when is even, our
approach is uniformly based on an phase space grid and thereby
avoids the necessity of having to invoke a `quadrupled' phase space and hence
the attendant redundance. Both odd and even cases are analysed in detail
and it is found that there are striking differences between the two. While the
odd case permits full implementation of the marginals property, the even
case does so only in a restricted sense. This has the consequence that in the
even case one is led to several equally good definitions of the Wigner
distributions as opposed to the odd case where the choice turns out to be
unique.Comment: Latex, 14 page
Congruences and Canonical Forms for a Positive Matrix: Application to the Schweinler-Wigner Extremum Principle
It is shown that a real symmetric [complex hermitian] positive
definite matrix is congruent to a diagonal matrix modulo a
pseudo-orthogonal [pseudo-unitary] matrix in [ ], for any
choice of partition . It is further shown that the method of proof in
this context can easily be adapted to obtain a rather simple proof of
Williamson's theorem which states that if is even then is congruent
also to a diagonal matrix modulo a symplectic matrix in
[]. Applications of these results considered include a
generalization of the Schweinler-Wigner method of `orthogonalization based on
an extremum principle' to construct pseudo-orthogonal and symplectic bases from
a given set of linearly independent vectors.Comment: 7 pages, latex, no figure
Classical Light Beams and Geometric Phases
We present a study of geometric phases in classical wave and polarisation
optics using the basic mathematical framework of quantum mechanics. Important
physical situations taken from scalar wave optics, pure polarisation optics,
and the behaviour of polarisation in the eikonal or ray limit of Maxwell's
equations in a transparent medium are considered. The case of a beam of light
whose propagation direction and polarisation state are both subject to change
is dealt with, attention being paid to the validity of Maxwell's equations at
all stages. Global topological aspects of the space of all propagation
directions are discussed using elementary group theoretical ideas, and the
effects on geometric phases are elucidated.Comment: 23 pages, 1 figur
Peres-Horodecki separability criterion for continuous variable systems
The Peres-Horodecki criterion of positivity under partial transpose is
studied in the context of separability of bipartite continuous variable states.
The partial transpose operation admits, in the continuous case, a geometric
interpretation as mirror reflection in phase space. This recognition leads to
uncertainty principles, stronger than the traditional ones, to be obeyed by all
separable states. For all bipartite Gaussian states, the Peres-Horodecki
criterion turns out to be necessary and sufficient condition for separability.Comment: 6 pages, no figure
The Sampling Theorem and Coherent State Systems in Quantum Mechanics
The well known Poisson Summation Formula is analysed from the perspective of
the coherent state systems associated with the Heisenberg--Weyl group. In
particular, it is shown that the Poisson summation formula may be viewed
abstractly as a relation between two sets of bases (Zak bases) arising as
simultaneous eigenvectors of two commuting unitary operators in which geometric
phase plays a key role. The Zak bases are shown to be interpretable as
generalised coherent state systems of the Heisenberg--Weyl group and this, in
turn, prompts analysis of the sampling theorem (an important and useful
consequence of the Poisson Summation Formula) and its extension from a coherent
state point of view leading to interesting results on properties of von Neumann
and finer lattices based on standard and generalised coherent state systems.Comment: 20 pages, Late
Gradient-based quantitative image reconstruction in ultrasound-modulated optical tomography: first harmonic measurement type in a linearised diffusion formulation
Ultrasound-modulated optical tomography is an emerging biomedical imaging
modality which uses the spatially localised acoustically-driven modulation of
coherent light as a probe of the structure and optical properties of biological
tissues. In this work we begin by providing an overview of forward modelling
methods, before deriving a linearised diffusion-style model which calculates
the first-harmonic modulated flux measured on the boundary of a given domain.
We derive and examine the correlation measurement density functions of the
model which describe the sensitivity of the modality to perturbations in the
optical parameters of interest. Finally, we employ said functions in the
development of an adjoint-assisted gradient based image reconstruction method,
which ameliorates the computational burden and memory requirements of a
traditional Newton-based optimisation approach. We validate our work by
performing reconstructions of optical absorption and scattering in two- and
three-dimensions using simulated measurements with 1% proportional Gaussian
noise, and demonstrate the successful recovery of the parameters to within
+/-5% of their true values when the resolution of the ultrasound raster probing
the domain is sufficient to delineate perturbing inclusions.Comment: 12 pages, 6 figure
Extragalactic H3O+: Some Consequences
We discuss some implications of our recent detection of extragalactic H3O+:
the location of the gas in M82, the origin of energetic radiation in M82, and
the possible feedback effects of star formation on the cosmic ray flux in
galaxies.Comment: Five pages, one figure; contribution to proceedings of conference
"Far-infrared observations of the interstellar medium", December 2007, Bad
Honne
Positive Lyapunov Exponents for Quasiperiodic Szego cocycles
In this paper we first obtain a formula of averaged Lyapunov exponents for
ergodic Szego cocycles via the Herman-Avila-Bochi formula. Then using
acceleration, we construct a class of analytic quasi-periodic Szego cocycles
with uniformly positive Lyapunov exponents. Finally, a simple application of
the main theorem in [Y] allows us to estimate the Lebesgue measure of support
of the measure associated to certain class of C1 quasiperiodic 2- sided
Verblunsky coefficients. Using the same method, we also recover the [S-S]
results for Schrodinger cocycles with nonconstant real analytic potentials and
obtain some nonuniform hyperbolicity results for arbitrarily fixed Brjuno
frequency and for certain C1 potentials.Comment: 27 papge
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